Estimating base sales volume using a low-pass filter approach

ABSTRACT

Within each iteration of an iterative process: (1) a low-pass filter is applied to an actual sales volumes series to extract low frequency components representing a base sales volume series for the iteration; and (2) a locally optimal base sales volume series is determined. A globally optimal base sales volume series is selected from among the locally optimal base sales volume series, comprising an estimated base sales volume for each time period. One or more of the estimated base sales volumes is made available for use in connection with at least one business analysis.

TECHNICAL FIELD OF THE INVENTION

This invention relates generally to business analysis and moreparticularly to estimating base sales volume using a low-pass filterapproach.

BACKGROUND OF THE INVENTION

Business analyses concerning products or other items are fundamentaltasks for many manufacturers, suppliers, retailers, and otherenterprises. Many business analyses are based on base sales volume. Forexample, demand forecasting, price-demand-behavior analysis, andpromotional effectiveness analysis all rely on sound base sales volumemeasurement. Base sales volume (sometimes referred to as base volume inthe remainder of this application) for consumer packed goods (CPG) istypically defined as the portion of the sales volume that would beexpected without advertising or other promotional support. In manybusiness analyses, base volume may be used as a de-causalized factor,such that a small difference in base volume may cause large deviationsin the consequent analysis. Many business analyses are frustrated bysuch propagation of errors in base volume estimation.

In general, base volume is an unobserved component of actual salesvolume (often referred to as actual volume in the remainder of thisapplication), which is the observed or recorded sales volume. Ideally,one could determine true base volume through observation, the accuracybeing limited only by the observational error. In practice, however, onecan only observe the actual volume—the true base volume cannot bedirectly observed. It is not difficult to show that any estimate of thetrue base volume based on the observed actual volume and a Lyapunoverror function (estimation criteria) will be biased. For example, themost common approach for determining promotional lifts based onestimated base volume generates incorrectly negatively signed lifts asmuch as thirty-five percent of the time; that is, determines a negativelift (decreased sales) when the lift should actually be positive(increased sales). In addition, conventional moving average approachesfor estimating base volume depend on both the number of leads and/orlags involved and weights associated with the leads and/or lags. Thecomplexity of the resulting space significantly impedes the ability tosearch for globally optimal solutions, making such approaches inferior.

Many businesses lack suitable approaches for measuring base salesvolume, for example, for CPG in the presence of one or more promotionalactivities. These businesses are therefore unable to fully understandthe various demand components for their products, which detracts fromtheir ability to effectively plan and manage important businessactivities. Furthermore, establishing effective approaches to thecomplex problem of estimating base volume has become an increasinglyimportant pursuit, from a theoretical perspective, in the academiccommunity. As a result of any of the above or other factors, priortechniques for estimating base volume have been inadequate to meet theneeds of many business and other users.

SUMMARY OF THE INVENTION

According to the present invention, disadvantages and problemsassociated with prior techniques for estimating base sales volume aresubstantially reduced or eliminated.

In one embodiment of the present invention, a method for estimating basesales volume includes accessing an input data series for a series oftime periods, the input data for each time period comprising at least anactual sales volume for the time period, the actual sales volumes forthe series of time periods collectively comprising an actual salesvolume series. Within each iteration of an iterative process: (1) alow-pass filter is applied to the actual sales volumes series in orderto extract low frequency components representing a base sales volumeseries for the iteration; and (2) a locally optimal base sales volumeseries is then determined for the iteration according to the input dataseries. A globally optimal base sales volume series is selected fromamong the locally optimal base sales volume series determined using theiterative process, the globally optimal base sales volume seriescomprising an estimated base sales volume for each time period. One ormore of the estimated base sales volumes is made available for use inconnection with at least one business analysis.

The present invention provides a number of important technicaladvantages over prior techniques. The present invention provides atechnique for extraction of unobservable base volume from the actualvolume through an iterative process that searches for globally optimalsolutions using a low-pass filter approach. Unlike previous techniquesthat depend on a large number of parameters and make global searchingessentially infeasible, the process of the present invention requiresonly a single smoothing parameter. The present invention also providesimproved criteria for selecting an optimal solution, which improves theaccuracy of the base volume estimation, particularly in the presence ofmoderate to high noise. Moreover, lift estimations according to thepresent invention are much less likely to be incorrectly signed thanwith previous techniques, for example, approximately ninety-five percentcorrectly positively signed compared to at most sixty-five percentcorrectly positively signed using previous techniques. Other importanttechnical advantages are readily apparent to those skilled in the artfrom the following figures, descriptions, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

To provide a more complete understanding of the present invention andthe features and advantages thereof, reference is made to the followingdescription taken in conjunction with the accompanying drawings, inwhich:

FIG. 1 illustrates an exemplary system for estimating base sales volumeusing a low-pass filter approach;

FIG. 2 illustrates an exemplary product dimension within amulti-dimensional database;

FIG. 3 illustrates an exemplary geography dimension within amulti-dimensional database;

FIG. 4 illustrates an exemplary method for estimating base sales volume;and

FIG. 5 illustrates an exemplary process for estimating base sales volumeusing a low-pass filter approach.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 illustrates an exemplary system 10 for estimating the base salesvolume using a low-pass filter approach. Although CPG or other suitableproducts are primarily referred to within this document, the presentinvention contemplates system 10 estimating base volume for suitabletangible or non-tangible items other than products, including but notlimited to services or other benefits. A particular application of thepresent invention is promotional planning, which may involve at least inpart the calculation of promotional “lifts” (increased sales volumeresulting from a promotional tactic), in view of any price change, foreach promotional tactic being evaluated. However, the present inventionmay be similarly applied to any other business analysis that depends onthe estimated base volume as input. Such analyses may include, forexample only and without limitation, demand forecasting, optimal markdown scheduling, complement analysis, and cannibalization analysis.

System 10 includes client 12, server 14, and a database 16. Client 12may include one or more processes to provide appropriate administration,analysis, and planning input. Although these processes are preferablyseparate processes running on a dedicated client processor, the presentinvention contemplates these processes being integrated, in whole or inpart, and running on one or more processors within the same or differentcomputers. Similarly, the server 14 may include one or more processes toreceive administration, analysis, and planning input from client 12 andinteract with database 16 to provide corresponding output to client 12.Although the processes are preferably separate processes running on adedicated server processor, the present invention contemplates theseprocesses being integrated, in whole or in part, and running on one ormore processors within the same or different computers. Client 12 andserver 14 may be fully autonomous or may operate at least in partsubject to input from users of system 10.

Database 16 provides persistent data storage for system 10. While theterm “database” is primarily used, a memory or other suitable datastorage arrangement may provide the functionality of database 16 withoutdeparting from the intended scope of the present invention. In oneembodiment, database 16 is hierarchical and multi-dimensional in nature.In a particular embodiment, the database 16 is three-dimensional andassociates with each storage location 18 a particular member of aproduct dimension, a particular member of a geography dimension, and aparticular member of a time dimension. Within database 16, eachcombination of members of these dimensions is associated with acorresponding storage location 18, similar to each combination ofcoordinates from the x-, y-, and z-axes being associated with a point inthree-dimensional Euclidean space. Furthermore, position in a particulardimension may be changed independent of members of other dimensions,like the position of a coordinate along the x-axis may be changedindependent of the positions of other coordinates along the y- andz-axes in three-dimensional Euclidean space. The values of one or moredata measures for a particular combination of members from the variousdimensions of the database 16 are stored in the particular storagelocation 18 for that combination of members.

The data measures associated with database 16 may include estimated basevolume and data measures from which the base volume is derived accordingto the present invention. In one embodiment, the other data measures mayinclude actual volume, incremental price reduction, one or morepromotion variables, or any other appropriate data measures. Actualvolume may be defined as the total volume sold over a specified timeperiod, expressed in suitable units (e.g., units, cases, pounds, etc.).Incremental price reduction may be defined as incremental pricereduction, expressed as a percentage of a base price or otherwise, as aresult of one or more promotional tactics during a specified timeperiod. Promotion variables may reflect promotional tactics that occurduring a specified time period (usually expressed in a binary format) orthe relative weights to be accorded those tactics. For example,promotion variables may reflect a temporary price reduction, insert,display, or any other suitable promotional tactic. If the tacticoccurred during the time period in question, the value of the variablemight equal one, otherwise the value might equal zero. The value of apromotion variable might instead be a weight, for example, to reflect apercentage of all commodity volume (% ACV), where ACV quantifies thesize of a store, chain, market, region, country, or the like in terms ofthe total sales. As a particular example, temporary price reduction %ACV might be a promotion variable reflecting the percentage of the totalsales for all stores that occur at stores which participated in sometemporary price reduction promotional tactic during the specified timeperiod.

The present invention contemplates database 16 having as few or as manydimensions as are appropriate in a particular case. For example, and notby way of limitation, an enterprise associated with system 10 may notconsider geography in connection with its base volume estimation needs.This might be the case where products are ordered using the Internet orthe telephone and then distributed from a single distribution point. Inthis example, the database 16 may be two-dimensional rather thanthree-dimensional and may not reflect positions or members within thegeography dimension. In another case, the values for base volume mayinherently reflect a time interval, in which case the database 16 may betwo-dimensional and may not reflect positions or members within the timedimension. Other scenarios involving more or fewer than three dimensionswill be readily apparent to those of skill in the art. The presentinvention contemplates database 16 having any suitable dimensions,according to the needs of the enterprise associated with system 10 andconstraints associated with the particular scenario.

In the three-dimensional embodiment of the present invention, the valuesof the data measures within the set for a particular storage location 18depend on the combined positions of members within product, geography,and time dimensions for that storage location 18. As a result, thevalues of the data measures typically vary with these combined positionsas appropriate to accurately reflect the actual volume, incrementalprice reduction, promotion variables, or other data associated with thecombination of members. As described below, when a combination ofmembers is specified in the product, geography, and time dimensionsaccording to operation of system 10, the database 16 accesses the valuesof the data measures for the storage location 18 associated with thatcombination of members for use in estimating base volume for thatcombination of members.

In one embodiment, the database 16 supports on-line analyticalprocessing (OLAP) capability and is populated with data measuresreceived from one or more transactional data sources that are internal,external, or both internal and external to the enterprise or facilityassociated with the system 10. For example and without limitation, datareceived from such sources may include, as described above, actualvolume, incremental price reduction, promotion variables, or any otherappropriate information that is applicable to estimation of base volume.The present invention contemplates data being stored in the database 16by server 14 based on input from client 12 or in any other suitablemanner.

Server 14 is coupled to database 16 using link 20, which may include anywireline, wireless, or other links suitable to support communicationsbetween server 14 and database 16 during operation of system 10.Database 16 may be integral to or separate from server 12, may operateon one or more computers, and may store any information suitable tosupport the operation of system 10 in estimating base volume accordingto the present invention. Client 12 is coupled to server 14 using link22, which may include any wireline, wireless, or other links supportingcommunications between client 12, server 14, and the processes of client12 and server 14 during operation of system 10. Although link 22 isshown generally coupling client 12 to server 14, the processes of server12 may communicate directly with corresponding processes of client 14according to particular needs.

In one embodiment, an administrator process of server 14 may communicatewith an administrator process of the client 12 to interact with anassociated user in managing database 16 and at least some activitiesassociated with the database 16. The administrator server process mayaccept user or other suitable input to define relationships, such asparent-child relationships, between the members of a single dimensionand between storage locations 18 that are each associated with multipledimensions within database 16. The administrator server process may alsoaccept user or other input to define the variation in time of the valuesof the data measures associated with a particular member. Theadministrator server process may store member relationships andvariability data within database 16 in any suitable format for use inestimating base volume according to the present invention.

An analyzer process of server 14 may communicate with an analyzerprocess of client 12 to access some or all of the contents of database16 for analysis and reporting purposes. For example, the analyzer serverprocess may provide analyzer client process read only access to database16, such that analyzer client process may not modify the contents ofdatabase 16.

A planner process of server 14 may communicate with a planner process ofclient 12 to allow client 12 or one or more associated users to storeinformation in database 16 and modify the contents of database 16 forpurposes of estimating base volume. For example, the planner serverprocess may participate in the storage and manipulation of dataexpressions and relationships that are used in estimating base volumeaccording to the present invention. The planner server process may alsoparticipate in modifying appropriate contents of database 16.

As described briefly above, client 12, server 14, and database 16 mayeach operate on one or more computers. Each computer may include one ormore input devices, such as a keypad, mouse, touch screen, microphone,or other device that receives information. Each computer may include anoutput device that conveys information associated with the operation ofsystem 10, including digital or analog data, visual information, oraudio information. Each computer may include fixed or removable storagemedia, such as a magnetic hard disk, CD-ROM, or other suitable storagemedia. Each computer may include one or more processors and associatedmemory to execute instructions and manipulate information during theoperation of system 10. Where multiple computers support client 12,server 14, and database 16, these computers may share one or moreresources as appropriate. Each of these one or more computers 34 may bea work station, personal computer, network computer, personal digitalassistant, wireless telephone, or any other suitable computing deviceaccording to particular needs.

FIG. 2 illustrates an exemplary product dimension 50 within database 16that includes a hierarchy of product levels 52 each having one or moremembers 54. The value of each data measure associated with a member 54is an aggregation of the values of corresponding data measuresassociated with hierarchically related members 54 in lower levels 52 ofproduct dimension 50. In an exemplary embodiment in which system 10provides estimated base volume, the base volume associated with a member54 is the aggregate estimated base volume for these hierarchicallyrelated members 54 in the lower levels 52 of the product dimension 50.In the illustrated embodiment, the product levels 52 for productdimension 50 include an all products level 58, a product type level 60,a product category level 62, and a product family level 64. Selectedexemplary hierarchical relationships between members 54 are shown usinglinks 56, as described below. Links 56 between hierarchically relatedmembers 54 in adjacent levels 52 of the product dimension 50 reflectparent-child relationships between members 54. Although FIG. 2 isdescribed primarily in connection with estimated base volumerelationships, the following description is similarly applicable toother relationships for other data measures, such as actual volume,incremental price reduction, promotion variables, or any otherappropriate data measures.

In the particular example shown in FIG. 2, all products level 58contains “All” member 54 representing the aggregate demand for allmembers 54 in lower levels 60, 62, and 64 of product dimension 50.Product type level 60 contains “Components,” “Base Units,” and “Options”members 54. “Components” member 54 represents the aggregate base volumefor hierarchically related members 54 below “Components” member 54 inlevels 62 and 64 of product dimension 50. Similarly, “Base Units” member54 represents the aggregate base volume for hierarchically relatedmembers 54 below “Base Units” member 54 and the “Options” member 54represents the aggregate base volume for hierarchically related members54 below “Options” member 54. Links 56 between “All” member 54 and“Components,” “Base Units,” and “Options” members 54 indicate thehierarchical relationships between these members 54.

Product category level 62 contains, under “Components” member 54, “HardDrives,” “Memory Boards,” and “CPUs” members 54. “Hard Drives” member 54represents the aggregate base volume for hierarchically related members54 below “Hard Drives” member 54 in level 64 of product dimension 50.Similarly, “Memory Boards” member 54 represents the aggregate basevolume for hierarchically related members 54 below “Memory Boards”member 54 and “CPUs” member 54 represents the aggregate base volume forhierarchically related members 54 below the “CPUs” member 54. The links56 between “Components” member 54 and “Hard Drives,” “Memory Boards,”and “CPUs” members 54 indicate the hierarchical relationships betweenthese members 54. Analogous links 56 reflect hierarchical relationshipsbetween “Base Units” and “Options” members 54 of the product type level60 and corresponding members 54 in lower levels 62 and 64 of productdimension 50.

Product family level 64 contains, under “Hard Drives” member 54, “4 GB”and “6 GB” members 54. Links 56 between “Hard Drives” member 54 and “4GB” and “6 GB” members 54 indicate hierarchical relationships betweenthese members 54. Analogous links 56 reflect hierarchical relationshipsbetween “Memory Boards,” “CPUs,” “Servers,” “Desktops,” “Laptops,”“Monitors,” “Keyboards,” and “Printers” members 54 of product categorylevel 62 and corresponding members 54 in lower level 64 within productdimension 50. Although no links 56 are shown between members 54 inproduct family level 64 and possible lower levels 52, the presentinvention contemplates further levels 52 existing within productdimension 50 and analogous links 56 to reflect correspondinghierarchical relationships. Furthermore, members 54 shown in FIG. 2 areexemplary only and are not intended to be an exhaustive set of all thepossible members 54. Those skilled in the art will readily appreciatethat other suitable members 54 and associated links 56 may exist withoutdeparting from the intended scope of the present invention.

FIG. 3 illustrates an exemplary geography dimension 70 within database16 that includes a hierarchy of geography levels 72 each with one ormore members 74. The value of each data measure associated with a member74 is an aggregation of the values of corresponding data measuresassociated with the hierarchically related members 74 in lower levels 72of the geography dimension 70. In the exemplary embodiment in whichsystem 10 provides estimated base volume, the base volume associatedwith a member 74 is the aggregate base volume for these hierarchicallyrelated members 74. In this embodiment, geography levels 72 for thegeography dimension 70 include a world level 78, a country level 80, aregion level 82, and a district level 84. Selected and merely exemplaryhierarchical relationships between members 74 are shown using links 76,which are analogous to links 56 described above with reference to FIG.2. Although FIG. 3 is described primarily in connection with estimatedbase volume relationships, the following description is similarlyapplicable to other relationships for other data measures, such asactual volume, percentage price reduction, promotional variables, or anyother appropriate data measures.

In the particular example illustrated in FIG. 3, world level 78 contains“World” member 74 representing aggregate worldwide base volume. Countrylevel 80 contains “U.S.” and “Canada” members 74, which represent theaggregate base volume for the United States and Canada, respectively.Link 76 between “U.S.” members 74 in the country level 80 and “World”members 74 in the world level 78 indicates a hierarchical relationshipbetween these members 74. Similarly, link 76 between “Canada” member 74and “World” member 74 indicates the hierarchical relationship betweenthese members 74. In this example, worldwide base volume is anaggregation of aggregate base volume in the United States as well asaggregate base volume in Canada. Although other links 76 are notdescribed in detail, those skilled in the art will appreciate that links76 are analogous to links 56 described above with reference to FIG. 2 inthat each link 56 represents a corresponding hierarchical relationshipbetween members 74 in various levels 72 of geography dimension 70. Asdescribed above, the present invention contemplates eliminating orotherwise not considering geography dimension 70 in estimating basevolume, for example, if geography dimension 70 is not relevant toparticular needs. Database 16 might in this situation betwo-dimensional.

Values for actual volume, incremental price reduction, promotionvariables, and any other data measures may be derived using traditionaltechniques. Deriving values for a particular member 54, 74 may depend onthe hierarchical relationships between the particular member 54, 74 andother members 54, 74. As is described above, the values associated witheach member 54, 74 are aggregations of values associated with members54, 74 in lower levels 52, 72 within the same hierarchy of parent-childlinks 56, 76. Therefore, given a value for a member 54, 74 (a parent) atone level 52, 72, the values for each of the related members 54 in thenext lowest level 52, 72 (children of the parent) may be determined bydisaggregating the value for the parent between its children. In oneembodiment, while the terms “parent” and “children” are used above toidentify a relationship between members 54, 74 of a single dimension 50,70, these terms may also be used to refer to the relationships betweenvalues associated with storage locations 18 that are each associatedwith members from multiple dimensions. For example, a first storagelocation 18 storing actual volume for a particular product in aparticular region may be hierarchically related to a second storagelocation 18 storing actual volume for the product in one city of thatregion (the value for the first storage location 18 being a parent ofthe value for the second storage location 18).

FIG. 4 illustrates an exemplary method for estimating base sales volume.The method begins at step 100, where server 14 receives actual volumeand any other suitable input data. For example, in addition to actualvolume data, input data may include data for incremental pricereduction, promotion variables, or other suitable data useful inestimating base volume according to the present invention. In oneembodiment, input data may be received as an input data series, withinput data for each of a succession of time periods, which might bedays, weeks, months, or any other appropriate time periods. For example,input data series may include actual volume series, incremental pricereduction series, promotion variable series, or any other suitable inputdata series. Server 14 may receive the input data from client 12 afterone or more users or other sources supply the input data to client 12.Server 14 stores the input data in database 16 at step 102.

At step 104, the server 14 may validate the input data for some or allstorage locations 18. In one embodiment, this involves validating theinput data for each intersection of members in the product and geographydimensions over the series in the time dimension. In a particularembodiment, validation involves determining a number of data points nwithin the input data series and determining the number of validindependent variables k, such as actual volume, incremental pricereduction, and promotion variables. For example, an independent variablemay be considered valid if the number of non-zero values in the timeseries for the independent variable is greater than one. Server 14 maykeep all input data series in their entirety for the intersections atissue, for example, where n≧k+5. Otherwise, portions of one or more suchseries may be discarded or otherwise ignored for purposes of estimatingbase volume. Although a particular validation technique is described,any suitable validation technique or no validation technique may be usedaccording to particular needs.

In general, the present invention applies a low-pass filter to theactual volume series for a particular intersection to extract the lowerfrequency components which represent a base volume series. The low-passfilter may be obtained as the solution to the following minimizationproblem:

$\begin{matrix}{\underset{{\{ s_{t}\}}_{t = 0}^{T + 1}}{Min}{\sum\limits_{t = 1}^{T}\;\left\{ \left\lbrack {\left( {y_{t} - s_{t}} \right)^{2} + {\lambda\left\lbrack {\left( {s_{t + 1} - s_{t}} \right) - \left( {s_{t} - s_{t - 1}} \right)} \right\rbrack}^{2}} \right\} \right.}} & (1)\end{matrix}$where y_(t) is an observed time series (usually non-stationary), s_(t)is a lower frequency component, and λ is a smoothing parameter withrange [0, +∞).

The closed form solution of problem (1) can be expressed as:LF=(λ·P+U)⁻¹  (2)where U is an identity matrix of dimension n (the number of observeddata points in the time series) and P is a Toeplitz matrix. To morefully understand the use of low-pass filtering in this context, considerthe limiting version of the first order closed form solution to problem(1):LF=[1+λ(1−B)²(1−B ⁻¹)²]⁻¹  (3)where B is backward operator.

The Fourier transform of this filter has a particularly simple form:

$\begin{matrix}{{{LF}(\omega)} = {\frac{1}{1 + {4{\lambda\left\lbrack {1 - {\cos(\omega)}} \right\rbrack}^{2}}}.}} & (4)\end{matrix}$Thus, the filter assigns a weight of approximately one at frequenciesclose to zero (since cos(0)=1) and assigns a weight of approximatelyzero at higher frequencies (since cos(π))=−1 implies thatLF(π)=1/(1+16λ), which is approximately zero for a large value of λ).

The bandwidth of this filter, and therefore its ability to filter outthe higher frequency components, depends only on the value of thesmoothing parameter, λ. Larger values of λ penalize changes in thelow-pass component and result in a smoother low-pass component. Thereliance on a single parameter, λ, as opposed to a very large number oftime and/or weight parameters as in previous techniques, significantlyreduces the complexity of the problem space. This makes searches forglobally optimal solutions feasible and provides an important technicaladvantage over such previous techniques.

At step 106, based on input from client 12 or otherwise, server 14 setsthe number I of values of the smoothing parameter λ that are to beiterated over when searching for global optima, and sets the number J ofiterations that are to occur for each λ in searching for local optima. Iand J are referred to for purposes of this document as iterationparameters. In general, I and J may be any suitable integers. However,to achieve a suitable balance of accuracy and computational speed, in apreferred embodiment server 14 sets I=5 and J=3. Letting λ(i) 10 ^(i−)1,equation (2) can be rewritten as:LF=(10^(i−1) ·P+U)⁻¹.  (5)In a case in which I=5, λ(i)=10^(i−1)=10,000, such that 10,000 values ofλ must be iterated over.

Server 14 accesses input data series stored within the database 16 atstep 108 and, at step 110, executes a process for estimating base volumethat implements the low-pass filter approach described above. Thisprocess is described in further detail below with reference to FIG. 5.After server 14 has selected an estimated base volume series accordingto this process, server 14 stores the estimated base volume values atthe appropriate storage locations 18 within the database 16 at step 112.At step 114, the estimated base volume values may be made available tothe server 14, client 12, an associated user, or another suitable entityfor use in connection with one or more business analyses, and the methodends.

FIG. 5 illustrates an exemplary process for estimating base sales volumeusing a low-pass filter approach, executed at step 110 of FIG. 4 as isdescribed above. Although a particular embodiment of the process will bedescribed, one or more appropriate additions, deletions, ormodifications may be made to the process without departing from theintended scope of the present invention.

The method begins at step 200, where server 14 initializes I specifyingthe values of the smoothing parameter λ to be iterated over:i=1.In general, as described briefly above, the process iterates over i insearching for a globally optimal base volume series. At step 202, theserver 14 creates a buffer to contain the temporary base volume series(TV) that is generated as a result of the calculations made during theiterative process:TV=AVwhere AV is the actual volume series. At step 204, server 14 initializesJ specifying number of iterations for each λ:j=1.In general, as briefly noted above, the process iterates over j insearching for locally optimal base volume series.

At step 206, server 14 applies an appropriate low-pass filter to extractlower frequency components, as described above:BV(i,j)=LF(i)*TV i=1,2, . . . , Iwhere BV is the base volume series for this iteration. At step 208,server 14 deletes or otherwise ignores non-promotion periods (keepingonly the data for time periods, referred to herein as promotion periods,during which at least one promotional tactic is used). At step 210,server 14 performs bias reduction for the remaining group of promotionperiods:BV(i,j)=AV if BV(i,j)>AV or BV(i,j)≦0.This or other bias reduction may be desirable to minimize leakage fromthe higher frequency components into the lower frequency components.

At step 212, server 14 calculates the value of a dependent variable y:

${y\left( {i,j} \right)} = {{\ln\left( \frac{AV}{{BV}\left( {i,j} \right)} \right)}.}$This serves as the left-hand side of the regression equation:y(i,j)=α+β₀ PR+β ₁ Prom ₁+ . . . +β_(k) Prom _(k)where PR is the incremental price reduction series (independentvariable) and each Prom is a corresponding promotion variable series(independent variable). At step 214, server 14 solves the regressionequation for the particular values of i,j for the iteration. A solutionof the regression equation is a set of estimated coefficients for theindependent variables. At step 216, the server 14 computes an errorcomponent associated with use of these estimated coefficients:e(i,j)=y(i,j)−ŷ(i,j)=y(i,j)−{circumflex over (α)}−{circumflex over (β)}₀PR−{circumflex over (β)} ₁ Prom ₁− . . . −{circumflex over (β)}_(k) Prom_(k).In one embodiment, at least approximately sixty-five percent (andpreferably approximately ninety-five percent) of the estimatedcoefficients generated according to the present invention are correctlypositively signed (indicating a sales increase when an increase shouldin fact occur).

At step 218, server 14 computes a new base volume series for theiteration:

${nBV} = {\frac{AV}{\exp\left( {e\left( {i,j} \right)} \right)}.}$where nBV is the new base volume series recovered. For each of the datapoints l in this series:

-   -   If nBV>BV₁(i,j), then nBV=BV₁(i,j) (for additional bias        reduction), else leave nBV as is (no bias reduction needed);    -   TV=nBV;    -   If at step 220 j<J set j=j+1, go to step 206, else go to step        222;    -   If at step 222 j=J and i<I set i=i+1, go to step 204, else go to        step 224.

At step 224, server 14 selects the globally optimal base volume series,BV*, from among the locally optimal base volume series (determined byiterating through all i,j) according to the following selectioncriteria:

-   -   If not all {circumflex over (β)}₀(i,j)<0 then BV*=BV(l,m) such        that

${{R^{2}\left( {l,m} \right)} = {\max\limits_{i,j}\left\{ {R^{2}\left( {i,j} \right)} \middle| {\forall{{{\hat{\beta}}_{0}\left( {i,j} \right)} > 0}} \right\}}};$

-   -   If all {circumflex over (β)}₀(i,j)<0 then BV*=BV(l,m) such that

${{\hat{\beta}}_{0}\left( {l,m} \right)} = {\max\limits_{i,j}{\left\{ {{\hat{\beta}}_{0}\left( {i,j} \right)} \middle| {\forall{{R^{2}\left( {i,j} \right)} > r}} \right\}.}}$where R² reflects, statistically, the “closeness” of the fit between theseries of actual values y(i,j) and estimated values ŷ(i,j), and r is avariable whose value is selected as an acceptable minimum value of R²according to particular business needs. In a particular embodiment, r isequal to 0.2. In essence, the first criterion eliminates base volumeseries with negative values for the estimated coefficient for theincremental price reduction series, {circumflex over (β)}₀. Then, forthe remaining base volume series (all having positive values of{circumflex over (β)}₀), the first criterion selects the base volumeseries with the value of {circumflex over (β)}₀ for which R² is maximum.In contrast, the second criterion in essence eliminates all base volumeseries for which R^(2>0.2). Then, from among the remaining base volumeseries, it selects the base volume series with the value of {circumflexover (β)}₀ that is least negative (closest to zero). These hybridselection criteria, which rely on a combination of sign-check rules andR² statistical rules, provide an important technical advantage overmodel selection criteria that rely entirely on R² statistical rules andare thus more likely to provide incorrect results in the presence ofmoderate to high noise.

The globally optimal base volume series, BV*, is the estimated basevolume series that may be used in connection with one or more businessanalyses. Once the values for estimated base volume have beendetermined, this process ends and the method of FIG. 4 continues.

Although the present invention has been described with severalembodiments, a plurality of changes, substitutions, variations,alterations, and modifications may be suggested to one skilled in theart, and it is intended that the invention encompass all such changes,substitutions, variations, alterations, and modifications as fall withinthe spirit and scope of the appended claims.

1. A computer-implemented method for estimating base sales volume, themethod being performed using one or more processing units, the methodcomprising: using one or more processing units, accessing an input dataseries for a series of time periods, the input data for each time periodcomprising at least an actual sales volume for the time period, theactual sales volumes for the series of time periods collectivelycomprising an actual sales volume series; within each iteration of aniterative process, wherein the iterative process involves a nonlinearregression process: using one or more processing units, applying alow-pass filter to the actual sales volumes series to extract lowfrequency components representing a base sales volume series for theiteration; using one or more processing units, determining a locallyoptimal base sales volume series for the iteration according to theinput data series; using one or more processing units, selecting aglobally optimal base sales volume series from among the locally optimalbase sales volume series determined using the iterative process, theglobally optimal base sales volume series comprising an estimated basesales volume for each time period; and using one or more processingunits, making one or more of the estimated base sales volumes availablefor use in connection with at least one business analysis, whereindetermining the locally optimal base sales volume series for aniteration comprises: solving a regression equation to determine valuesfor estimated coefficients associated with an incremental pricereduction series and one or more promotion variable series; computing anerror associated with use of the estimated coefficients; and to reducebias, considering the error in selecting the locally optimal base salesvolume series.
 2. The method of claim 1, wherein the iterative processis performed according to a smoothing parameter that is independent oftime periods associated with the input data series.
 3. The method ofclaim 1, wherein: a first parameter specifies the number of values thesmoothing parameter can have, an iterative loop being performed withinthe iterative process for each value of the smoothing parameter; and asecond parameter specifies the number of iterations to be performed,inside the iterative loop, for each value of the smoothing parameter. 4.The method of claim 3, wherein the smoothing parameter will haveapproximately ten thousand values according to the first parameter and,according to the second parameter, approximately three iterations willbe performed inside the iterative loop for each value of the smoothingparameter.
 5. The method of claim 1, wherein: the input data series isstored in a multi-dimensional database comprising at least product,geography, and time dimensions; and each input data value in the inputdata series is associated with a particular intersection of memberswithin the product, geography, and time dimensions.
 6. The method ofclaim 1, wherein the input data for each time period further comprises:an incremental price reduction value associated with one or morepromotional tactics conducted in the time period, the incremental pricereduction values for the time periods collectively comprising anincremental price reduction series; and values for one or more promotionvariables that reflect whether associated promotional tactics areconducted during the time period or reflect relative weights accordedassociated promotional tactics conducted during the time period, thevalues for each promotion variable collectively comprising a promotionvariable series for that promotion variable, one or more of thesepromotional tactics selected from the group consisting of: a temporaryprice reduction for the item; a promotional insert packaged with theitem; a promotional display for the item; and an advertisement for theitem.
 7. The method of claim 1, wherein the regression equation involvesthe actual sales volume series, the base sales volume series resultingfrom application of the low-pass filter, the incremental price reductionseries, and the one or more promotion variable series.
 8. The method ofclaim 7, wherein selecting the globally optimal base sales volume seriescomprises: eliminating all the locally optimal base sales volume serieshaving negative values for the estimated coefficient for the incrementalprice reduction series; and of the remaining locally optimal base salesvolume series, selecting the locally optimal base sales volume seriesfor which an R² statistical measure has a maximum value.
 9. The methodof claim 7, wherein selecting the globally optimal base sales volumeseries comprises: eliminating all the locally optimal base sales volumeseries for which an R² statistical measure has a value less thanapproximately 0.2; and of the remaining locally optimal base salesvolume series, selecting the locally optimal base sales volume serieshaving the estimated coefficient for the incremental price reductionseries with least negative value.
 10. The method of claim 7, whereinless than approximately thirty-five percent of the values of theestimated coefficients have incorrect signs indicating a decrease inbase sales volume when in reality an increase should occur.
 11. Themethod of claim 7, wherein approximately ninety-five percent of thevalues of the estimated coefficients have correct signs indicating anincrease in base sales volume when in reality an increase should occur.12. The method of claim 1, wherein the business analysis comprises thecalculation of increased sales volume associated with a promotionaltactic based on one or more estimated base sales volumes.
 13. The methodof claim 1, wherein the business analysis is selected from the groupconsisting of: promotional planning; demand forecasting; optimal markdown scheduling; complement analysis; and cannibalization analysis. 14.A system for estimating base sales volume, comprising: a databaseoperable to store an input data series for a series of time periods, theinput data for each time period comprising at least an actual salesvolume for the time period, the actual sales volumes for the series oftime periods collectively comprising an actual sales volume series; oneor more processors collectively operable to: access the input dataseries; within each iteration of an iterative process that involves anonlinear regression process: apply a low-pass filter to the actualsales volumes series to extract low frequency components representing abase sales volume series for the iteration; determine a locally optimalbase sales volume series for the iteration according to the input dataseries; select a globally optimal base sales volume series from amongthe locally optimal base sales volume series determined using theiterative process, the globally optimal base sales volume seriescomprising an estimated base sales volume for each time period; and makeone or more of the estimated base sales volumes available for use inconnection with at least one business analysis, wherein determining thelocally optimal base sales volume series for an iteration comprises:solving a regression equation to determine values for estimatedcoefficients associated with an incremental price reduction series andone or more promotion variable series; computing an error associatedwith use of the estimated coefficients; and to reduce bias, consideringthe error in selecting the locally optimal base sales volume series. 15.The system of claim 14, wherein the processor performs the iterativeprocess according to a smoothing parameter which is independent of timeperiods associated with the input data series.
 16. The system of claim14, wherein: a first parameter specifies the number of values thesmoothing parameter can have, an iterative loop being performed withinthe iterative process for each value of the smoothing parameter; and asecond parameter specifies the number of iterations to be performed,inside the iterative loop, for each value of the smoothing parameter.17. The system of claim 16, wherein the smoothing parameter will haveapproximately ten thousand values according to the first parameter and,according to the second parameter, approximately three iterations willbe performed inside the iterative loop for each value of the smoothingparameter.
 18. The system of claim 14, wherein: the database ismulti-dimensional and comprises at least product, geography, and timedimensions; and each input data value in the input data series isassociated with a particular intersection of members within the product,geography, and time dimensions.
 19. The system of claim 14, wherein theinput data for each time period further comprises: an incremental pricereduction value associated with one or more promotional tacticsconducted in the time period, the incremental price reduction values forthe time periods collectively comprising an incremental price reductionseries; and values for one or more promotion variables that reflectwhether associated promotional tactics are conducted during the timeperiod or reflect relative weights accorded associated promotionaltactics conducted during the time period, the values for each promotionvariable collectively comprising a promotion variable series for thatpromotion variable, one or more of these promotional tactics selectedfrom the group consisting of: a temporary price reduction for the item;a promotional insert packaged with the item; a promotional display forthe item; and an advertisement for the item.
 20. The system of claim 14,wherein the regression equation involves the actual sales volume series,the base sales volume series resulting from application of the low-passfilter, the incremental price reduction series, and the one or morepromotion variable series.
 21. The system of claim 20, wherein selectingthe globally optimal base sales volume series comprises: eliminating allthe locally optimal base sales volume series having negative values forthe estimated coefficient for the incremental price reduction series;and of the remaining locally optimal base sales volume series, selectingthe locally optimal base sales volume series for which an R² statisticalmeasure has a maximum value.
 22. The system of claim 20, whereinselecting the globally optimal base sales volume series comprises:eliminating all the locally optimal base sales volume series for whichan R² statistical measure has a value less than approximately 0.2; andof the remaining locally optimal base sales volume series, selecting thelocally optimal base sales volume series having the estimatedcoefficient for the incremental price reduction series with leastnegative value.
 23. The system of claim 20, wherein less thanapproximately thirty-five percent of the values of the estimatedcoefficients have incorrect signs indicating a decrease in base salesvolume when in reality an increase should occur.
 24. The system of claim20, wherein approximately ninety-five percent of the values of theestimated coefficients have correct signs indicating an increase in basesales volume when in reality an increase should occur.
 25. The system ofclaim 14, wherein the business analysis comprises the calculation ofincreased sales volume associated with a promotional tactic based on oneor more estimated base sales volumes.
 26. The system of claim 14, hereinthe business analysis is selected from the group consisting of:promotional planning; demand forecasting; optimal mark down scheduling;complement analysis; and cannibalization analysis.
 27. Software forestimating base sales volume, the software embodied in acomputer-readable medium and when executed by a computer operable to:access an input data series for a series of time periods, the input datafor each time period comprising at least an actual sales volume for thetime period, the actual sales volumes for the series of time periodscollectively comprising an actual sales volume series; within eachiteration of an iterative process that involves a nonlinear regressionprocess: apply a low-pass filter to the actual sales volumes series toextract low frequency components representing a base sales volume seriesfor the iteration; determine a locally optimal base sales volume seriesfor the iteration according to the input data series; select a globallyoptimal base sales volume series from among the locally optimal basesales volume series determined using the iterative process, the globallyoptimal base sales volume series comprising an estimated base salesvolume for each time period; and make one or more of the estimated basesales volumes available for use in connection with at least one businessanalysis, wherein determining the locally optimal base sales volumeseries for an iteration further comprises: solving a regression equationto determine values for estimated coefficients associated with anincremental price reduction series and one or more promotion variableseries; computing an error associated with use of the estimatedcoefficients; and to reduce bias, considering the error in selecting thelocally optimal base sales volume series.
 28. The software of claim 27,wherein the iterative process is performed according to a smoothingparameter which is independent of time periods associated with the inputdata series.
 29. The software of claim 27, wherein: a first parameterspecifies the number of values the smoothing parameter can have, aniterative loop being performed within the iterative process for eachvalue of the smoothing parameter; and second parameter specifies thenumber of iterations to be performed, inside the iterative loop, foreach value of the smoothing parameter.
 30. The software of claim 29,wherein the smoothing parameter will have approximately ten thousandvalues according to the first parameter and, according to the secondparameter, approximately three iterations will be performed inside theiterative loop for each value of the smoothing parameter.
 31. Thesoftware of claim 27, wherein: the input data is stored in amultidimensional database comprising at least product, geography, andtime dimensions; and each input data value in the input data series isassociated with a particular intersection of members within the product,geography, and time dimensions.
 32. The software of claim 27, whereinthe input data for each time period further comprises: an incrementalprice reduction value associated with one or more promotional tacticsconducted in the time period, the incremental price reduction values forthe time periods collectively comprising an incremental price reductionseries; and values for one or more promotion variables that reflectwhether associated promotional tactics are conducted during the timeperiod or reflect relative weights accorded associated promotionaltactics conducted during the time period, the values for each promotionvariable collectively comprising a promotion variable series for thatpromotion variable, one or more of these promotional tactics selectedfrom the group consisting of: a temporary price reduction for the item;a promotional insert packaged with the item; a promotional display forthe item; and an advertisement for the item.
 33. The software of claim27, wherein the regression equation involves the actual sales volumeseries, the base sales volume series resulting from application of thelow-pass filter, the incremental price reduction series, and the one ormore promotion variable series.
 34. The software of claim 33, whereinselecting the globally optimal base sales volume series comprises:eliminating all the locally optimal base sales volume series havingnegative values for the estimated coefficient for the incremental pricereduction series; and of the remaining locally optimal base ales volumeseries, selecting the locally optimal base sales volume series for whichan R statistical measure has a maximum value.
 35. The software of claim33, wherein selecting the globally optimal base sales volume seriescomprises: eliminating all the locally optimal base sales volume seriesfor which an statistical measure has a value less than approximately0.2; and of the remaining locally optimal base sales volume series,selecting the locally optimal base sales volume series having theestimated coefficient for the incremental price reduction series withleast negative value.
 36. The software of claim 33, wherein less thanapproximately thirty-five percent of the values of the estimatedcoefficients have incorrect signs indicating a decrease in base salesvolume when in reality an increase should occur.
 37. The software ofclaim 33, wherein approximately ninety-five percent of the values of theestimated coefficients have correct signs indicating an increase in basesales volume when in reality an increase should occur.
 38. The softwareof claim 27, wherein the business analysis comprises the calculation ofincreased sales volume associated with a promotional tactic based on oneor more estimated base sales volumes.
 39. The software of claim 27,wherein the business analysis is selected from the group consisting of:promotional planning; demand forecasting; optimal mark down scheduling;complement analysis; and cannibalization analysis.
 40. A system forestimating base sales volume, comprising: data storage means for storingan input data series for a series of time periods, the input data foreach time period comprising at least an actual sales volume for the timeperiod, the actual sales volumes for the series of time periodscollectively comprising an actual sales volume series; and processingmeans for: accessing the input data series; within each iteration of aniterative process that involves a nonlinear regression process; applyinga low-pass filter to the actual sales volumes series to extract lowfrequency components representing a base sales volume series for theiteration; determining a locally optimal base sales volume series forthe iteration according to the input data series; selecting a globallyoptimal base sales volume series from among the locally optimal basesales volume series determined using the iterative process, the globallyoptimal base sales volume series comprising an estimated base salesvolume for each time period; and making one or more of the estimatedbase sales volumes available for use in connection with at least onebusiness analysis, wherein determining the locally optimal base salesvolume series for an iteration further comprises: solving a regressionequation to determine values for estimated coefficients associated withan incremental price reduction series and one or more promotion variableseries; computing an error associated with use of the estimatedcoefficients; and to reduce bias, considering the error in selecting thelocally optimal base sales volume series.